Journal of Applied Probability

Critical scaling for the SIS stochastic epidemic

R. G. Dolgoarshinnykh and Steven P. Lalley

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We exhibit a scaling law for the critical SIS stochastic epidemic. If at time 0 the population consists of √N infected and N - √N susceptible individuals, then when the time and the number currently infected are both scaled by √N, the resulting process converges, as N → ∞, to a diffusion process related to the Feller diffusion by a change of drift. As a consequence, the rescaled size of the epidemic has a limit law that coincides with that of a first passage time for the standard Ornstein-Uhlenbeck process. These results are the analogs for the SIS epidemic of results of Martin-Löf (1998) and Aldous (1997) for the simple SIR epidemic.

Article information

J. Appl. Probab. Volume 43, Number 3 (2006), 892-898.

First available in Project Euclid: 20 September 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx]
Secondary: 92D30: Epidemiology

Stochastic epidemic model SIS SIR Feller diffusion Ornstein-Uhlenbeck process


Dolgoarshinnykh, R. G.; Lalley, Steven P. Critical scaling for the SIS stochastic epidemic. J. Appl. Probab. 43 (2006), no. 3, 892--898. doi:10.1239/jap/1158784956.

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  • Aldous, D. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Prob. 25, 812--854.
  • Darling, D. A. and Siegert, A. J. F. (1953). The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624--639.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. John Wiley, New York.
  • Feller, W. (1951). Diffusion processes in genetics. In Proc. 2nd Berkeley Symp. Math. Statist. Prob., 1950, University of California Press, Berkeley and Los Angeles, pp. 227--246.
  • Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ.
  • Jiřina, M. (1969). On Feller's branching diffusion processes. Časopis P\v est. Mat. 94, 84--90, 107.
  • Kryscio, R. J. and Lefèvre, C. (1989). On the extinction of the S-I-S stochastic logistic epidemic. J. Appl. Prob. 26, 685--694.
  • Lindvall, T. (1974). On Feller's branching diffusion processes. Adv. Appl. Prob. 6, 309--321.
  • Martin-Löf, A. (1998). The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier. J. Appl. Prob. 35, 671--682.
  • Nasell, I. (1996). The quasi-stationary distribution of the closed endemic SIS model. Adv. Appl. Prob. 28, 895--932.
  • Nasell, I. (1999). On the time to extinction in recurrent epidemics. J. R. Statist. Soc. B 61, 309--330.
  • Norden, R. H. (1982). On the distribution of the time to extinction in the stochastic logistic population model. Adv. Appl. Prob. 14, 687--708.
  • Weiss, G. and Dishon, M. (1971). On the asymptotic behavior of the stochastic and deterministic models of an epidemic. Math. Biosci. 11, 261--265.